Maximum likelihood estimation for multivariate normal distribution with monotone sample

Estimation of a Multivariate Normal Covariance Matrix with Staircase Pattern Data

“…For this pattern, Liu (1993) presents a decomposition of the posterior distribution of under a family of prior distributions. Jinadasa and Tracy (1992) obtain a complicated form for the maximum likelihood estimators of the unknown mean and the covariance matrix in terms of some sufficient statistics, which extends the work of Anderson and Olkin (1985). Recently, Kibria, Sun, Zidek and Le (2002) discussed estimating using a generalized inverted Wishart (GIW) prior and applied the result to mapping PM 2.5 exposure.…”

“…In Sect. 3, we consider the Cholesky decomposition of and derive a closed form expression of the MLE of based on a set of sufficient statistics different from those in Jinadasa and Tracy (1992). We also show that the best equivariant estimator of with respect to the lower-triangular matrix group uniquely exists under the Stein invariant loss function, resulting in the inadmissibility of the MLE.…”

Efficient ML Estimation of the Multivariate Normal Distribution from Incomplete Data

“…In other words, the ML estimates of the parameters are obtained in closed form. This result is not new (see, e.g., Anderson, 1959;Little and Rubin, 1987;Jinadasa and Tracy, 1992). With the extension of Bartlett's decomposition to monotone samples, the ML estimates have neat closed-form expressions and the associated Fisher information matrix can also be expressed in closed form.…”

“…A way of finding the MLEs of parameters given one of those patterns is to use a general method called the factored likelihood approach (Little and Rubin, 1987). The closed-form expression of the MLEs of the parameters with a monotone missing-data pattern appears in different places; a recent example is Jinadasa and Tracy (1992). For an incomplete normal dataset with a general missing-data pattern, Hocking and Smith (1968) provided a sequential method for estimation of the multivariate normal distribution.…”

The maximum likelihood estimators in a multivariate normal distribution with AR(1) covariance structure for monotone data

“…Assume that monotone data are independently distributed as Np (D,F~), in other words, x~j's are mutually independent and ~j is distributed as Ni (Di, Ei) where It i is the first i components of # and Ei is the first i × i matrix of E. The maximum likelihood estimators of # and E were investigated by Anderson (1957), Bhargava (1975, Jinadasa and Tracy (1992) and Fujisawa (1995)• It is sometimes postulated that ~ has a structure given by E ----cr2(pli-jl), which is called AR(1) covariance structure. An example is the case when data is longitudinal.…”